Atlas › Modular forms › Weight 8 › Level 50 › 50.8.a › 50.8.a.j

Newform orbit 50.8.a.j

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 16q^{2} + 56q^{3} + 128q^{4} + 448q^{6} + 408q^{7} + 1024q^{8} + 3394q^{9} + 8904q^{11} + 3584q^{12} - 3504q^{13} + 3264q^{14} + 8192q^{16} + 43648q^{17} + 27152q^{18} - 31800q^{19} - 31976q^{21} + 71232q^{22} + 71656q^{23} + 28672q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{31})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 464x - 57776$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	— ( $5^2 \mid 50$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	16	56	0	408	8904	−3504	43648	−31800	71656	−84780	−197056	195408	116244	275736	−971272	−2007504	−1046760	−2625716	1678728	−3916176	2606496	−3863520	5183416	16735020	12604608

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 5, 10, 25, 50]: [1, 3, 5, 13, 19, 55] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(50, 8)[9]   # 50.8.a.j

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 50:8 1000